Arithmetic+Sequences+2C-1

= = **__Arithmetic Sequences__**
 * 1) An Arithmetic Sequence is a sequence of numbers that increases by the same amount every time. toc
 * 2) The formula for an Arithmetic Sequence is F(x) = mx + b.
 * 3) When you graph an Arithmetic Sequence the graph will always be a straight line.

__ Examples __
**Example #1** (0, 5, 10, 15) is an example is of a sequence that starts at 0 and increases by 5 each time. The formula for this sequence is F(x) = 5x + 0. (2, 6, 10, 14) is an example is of a sequence that has a starting number of 2 and increases by 4 each time. The formula for this sequence is F(x) = 4x + 2.
 * Example #2**

__ Understanding the formula __
math F(x)= mx + b math

Where "m" represents the amount each number in the sequence increases by each time "x" represents the term in the sequence that you're trying to find. "b" represents the starting point also called the 0th term of the sequence.

__**Example #1 continued**__ (0, 5, 10, 15) is an example is of an arithmetic sequence. For this sequence the starting point called 'b' is 0 and each term increases by 5 which is called 'm'. Now that we know these two pieces of information we can write a formula find the value for the xth term. The equation would be

math F(x)= 5x + 0 math

To make the construction of our formula easier to understand and to see why it works, we can put the sequence into a chart.
 * > xth term ||> ** 0 **th (starting term) ||> ** 1 **st term ||> ** 2 **nd term ||> ** 3 **rd term ||
 * = F(x) ||= 0 ||= 5 ||= 10 ||= 15 ||


 * > xth term ||> ** 0 **th (starting term) ||> ** 1 **st term ||> ** 2 **nd term ||> ** 3 **rd term ||
 * = F(x) ||= 0 or ( ** 0 ** x 5) ||= 5 or ( ** 1 ** x 5) ||= 10 or ( ** 2 ** x 5) ||= 15 or ( ** 3 ** x 5) ||

Now that we know our formula works we can use it to find out to find out what the 16th term of the sequence would be. All you have to do is plug in 16 for x and then solve. When we do, solve we see that

math F(16)= 5(16) + 0 = 80 math

So the 16th term in the sequence would be 80.

__ Further Discussion __

 * What is the purpose of constructing a formula?
 * What are some connections between the graph, table, and formula for the arithmetic sequence is graph #1?
 * How does Y=mx +b (linear equations/ slope-intercept form) relate to arithmetic sequences?